This isn't quite accurate.
It is accurate to say that the Union Jack is a superposition of three flags:
At the time of the Act of Union, Ireland did not have a national flag. Insofar as anything at that time could be called a national flag, it would be the Green Flag. The Cross of St. Patrick was created based on the crest of the Order of St. Patrick, which was founded in the 1780's or so (reasonably close to the Act of Union), but it's inaccurate to say that it was "the flag of Ireland." -- EdwardOConnor